2010/06/28 by Daniel V. Mathews, Mathews, Daniel V.
Mathematics · #57M50 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1006.5384
openalex publication_date 2010/06/28 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
We consider the relationship between hyperbolic cone-manifold structures on\nsurfaces, and algebraic representations of the fundamental group into a group\nof isometries. A hyperbolic cone-manifold structure on a surface, with all\ninterior cone angles being integer multiples of 2\π, determines a holonomy\nrepresentation of the fundamental group. We ask, conversely, when a\nrepresentation of the fundamental group is the holonomy of a hyperbolic\ncone-manifold structure. In this paper we build upon previous work with\npunctured tori to prove results for higher genus surfaces. Our techniques\nconstruct fundamental domains for hyperbolic cone-manifold structures, from the\ngeometry of a representation. Central to these techniques are the Euler class\nof a representation, the group widetildePSL2 R, the twist of hyperbolic\nisometries, and character varieties. We consider the action of the outer\nautomorphism and related groups on the character variety, which is\nmeasure-preserving with respect to a natural measure derived from its\nsymplectic structure, and ergodic in certain regions. Under various hypotheses,\nwe almost surely or surely obtain a hyperbolic cone-manifold structure with\nprescribed holonomy.\n